A Dynamic Risk Measure

The Iterated CTE – a Dynamic Risk Measure

Mary R. Hardy Julia L. Wirch Dept of Statistics and Dept of Actuarial Mathematics Actuarial Science and Statistics University of Waterloo Heriot-Watt University Ontario, Canada Edinburgh, Scotland [email protected] [email protected]

Abstract

In this paper we present a method for defining a dynamic risk measure from a static risk measure, by backwards iteration. We apply the method to the CTE risk measure to construct a new, dynamic risk measure, the iterated CTE or ICTE. We show that the ICTE is coherent, consistent and relevant according to the defini- tions of Riedel (2003), and we derive formulae for the ICTE for the case where the loss process is lognormal. Finally, we demonstrate the practical implementation of the ICTE to an equity-linked insurance contract with maturity and death benefit guarantees.

1 1 Introduction

1.1 Quantile and CTE risk measures

A risk measure maps a loss or profit random variable to the real line. The outcome is a quantification of the risk associated with a random loss. We commonly use risk measures in actuarial science to determine premiums or capital requirements for an uncertain future liability. The quantile risk measure is a traditional actuarial science measure, used both for capital requirements and for premiums. This has been separately adopted by the banking indus- try as ‘Value at Risk’ or VaR. The quantile risk measure with parameter α, 0 ≤ α < 1 is the α-quantile of the loss distribution. If the loss random variable is X > 0, then the α-quantile random variable is denoted here by Qα(X); that is Pr[X ≤ Qα(X)] = α for a continuous loss random variable X. Using this measure for capital requirements is interpreted as setting capital requirements with a probability α that the capital will be sufficient to meet the liability. Various authors have identified problems with the quantile approach. It is not coherent, in the sense of Artzner et al (2002); it can give perverse capital requirements, and it can be manipulated to mask substantial risks. It may encourage the sale of low frequency, high severity risks. Several authors have noted these. For further details of the theoretical and practical problems with the quantile or VaR approach see Artzner et al (1999), Boyle and Vorst (2003), and Wirch and Hardy (1999). The Conditional Tail Expectation (CTE) approach does not have these disadvantages. The CTE risk measure applied at some standard α, where 0 ≤ α < 1 as before, for a random loss X which is continuous around the quantile value Qα(X), is

Cα(X) = E[X|X > Qα(X)] (1)

That is, the capital required is suﬃcient to meet the loss in the event of the worst (1 − α)

2 event, on average. This equation for the CTE needs adjustment if Qα(X) falls in a probability mass. In the more general case the CTE with parameter α is calculated as follows. Find β0 = max{β : Qα(X) = Qβ(X)} (1 − β0) E[X|X > Q ] + (β0 − α) Q then C (X) = α α (2) α 1 − α Another way of allowing for the probability mass problem is to use a distortion approach:   y if 0 ≤ y < 1 − α Let g(y) = 1−α (3)  1 if 1 − α ≤ y ≤ 1 and let SX (x) denote the decumulative distribution function of X, then Z ∞ Cα(X) = g(SX (x)) dx (4) 0 The CTE risk measure (also called Tail-VaR or expected shortfall) is coherent. It is also quite intuitive and is already widely used in actuarial science. The CTE risk measure has been adopted by the Canadian Institute of Actuaries in its recommendations with respect to the risk capital requirements for equity-linked contracts (SFTF (2002)), and is also widely used in the USA (see, for example, O’Connor (2002)).

1.2 The multi-period problem

The quantile and CTE risk measures are static. That is, there is no explicit allowance for the risk measure to evolve as time progresses. A risk measure which is deﬁned over a process rather than for a ﬁxed liability is called a multi-period, or dynamic risk measure. The problem of dynamic or multi-period risk measures has been gaining much attention recently. Suppose an insurer has a liability due in 20 years. The static approach to risk measurement would give a value for the economic or regulatory capital to be held at the valuation date,

3 taking into consideration the distribution of outcomes for the future loss. However, the risk manager may be more immediately interested in having sufficient funds to meet the economic capital requirements in the following year, and the year after that and so on, as well as meeting the final liability. He or she will also be concerned not to hold excessive capital if the risk evolves favorably to the company. So a multi-period risk measure might allow not only the final liability but also for the intermediate capital requirements, taking into consideration how the loss process evolves. It should also allow for liability cashflows over time, as well as the single future liability. Artzner et al.(2003), Wang(2000) and Riedel(2003)have explored the multi-period problem using slightly different axiomatic approaches. The key to defining their risk measures is determining the appropriate closed convex set of test probabilities (scenarios) over which to calculate expected values. The test probabilities could be defined using a stopping-time process which will then allow any intermediate portfolio value to contribute to the risk measure calculation. Defining this set of test probabilities is the most difficult hurdle to implementing the multi-period risk measure. In this paper we take a more applied approach. Rather than start with axioms, we start with a single period risk measure; extend it to a multi-period framework, and then explore the characteristics of the dynamic risk measure. Our approach to the multi-period extension of the single period problem may be applied to any static risk measure. We focus on the CTE measure, as it is well known, coherent and commonly used for equity-linked insurance. In the risk measure literature there are commonly two different types of risk considered:

• The rolling horizon portfolio: in the application of Value at Risk to banking, typi- cally the risk is the potential portfolio loss over the following 10 days. Each day the risk is re-assessed with a new horizon.

• The fixed horizon portfolio; for example, where the risk measure is applied to a specific liability that expires at the fixed horizon. This would include the application of risk measures to a single equity-linked insurance contract, or a single cohort of

4 contracts. Where a whole portfolio of contracts with multiple maturity dates is to be risk measured, a ﬁxed horizon (perhaps the ﬁnal maturity) may be used, but a rolling horizon is also possible, particularly if new business is included.

In this paper we are interested in the ﬁxed horizon case, particularly as applied to the economic or regulatory capital for equity-linked contracts of a class which includes variable annuities and segregated funds. However, we believe that the method is easily adapted to the rolling horizon case.

1.3 Outline

In the following section we use a simple binomial example to motivate the multi-period approach and to describe, intuitively, our solution. In Section 3 we deﬁne the iterated risk measure slightly more formally, and explore the characteristics of the measure. The rest of the paper considers two worked examples of the iterated CTE. In Section 4, we derive analytic results for a lognormal loss process. In Section 5 we work through an example for a segregated fund insurance contract1.

2 Example

Consider the liability represented in the following 2-step model.

1Segregated fund contracts are known as Variable annuities in the USA and as Unit Linked contracts elsewhere.

5 100 0.06 (1,1) P ¨ PP p = 0.06¨¨ PP ¨ PP ¨ PP 0 ¨¨ 0.94 ¨¨ (0) H HH HH 50 H 0.06 H HH 1 − p = 0.94 (1,2) P PP PP PP 0.94 PP 0

At the start of the process, that is at node (0), the ﬁnal liability X, due at time 2, takes values:    100 with probability 0.0036 X = 50 with probability 0.0564   0 with probability 0.94

Suppose the regulator requires a 95% CTE to be demonstrated at each time point. At the start of the contract the 95% CTE of the ﬁnal liability is: (0.0036)(100) + (0.05 − 0.0036)(50) = 53.60 0.05 After one time unit the process is either at node (1,1) or node (1,2). At node (1,1), the 95% CTE to maturity is $100 . At node (1,2) the 95% CTE is $50. Note that the CTE held at time zero will be inadequate by $46.40 at time 1 if the process moves to node (1,1), which has 6% probability. So the static approach ignores a substantial probability of an additional draw on funds after the ﬁrst time unit, despite the fact that the additional capital requirement is completely predictable in probability.

For a multi-period risk measure we need a process Jt,T (X) where X is the future loss, t

6 is the time variable, T is the date at which the liability X matures. The simplest approach is to re-apply the single period risk measure at each time period. We call this approach the re-calculated CTE, since at each time unit we re-calculate the CTE to maturity. Another possibility, supposing the regulator is not involved, is simply to hold the initial risk measure capital of $53.60 through to maturity. This has the advantage that no additional draw on funds is required at time 1. However, if at time 1 the process is at node (1,2), then risk measure of $53.60 is being held in respect of a liability of at most $50.00. This approach therefore may introduce excessive capital requirements. We call this approach the accumulated CTE, since we simply hold the initial CTE, with interest if appropriate (we have ignored interest in this example). These two examples lead to an obvious candidate for a multi-period risk measure. Suppose at each time unit, the random liability considered for our risk measure is not the ﬁnal liability, but instead is the potential risk measure required in the following time period. In terms of the example above, the CTE at time 0 would relate to the potential risk capital required at time 1 – that is $100 at node (1,1) and $50 at node (1,2). At node (0) we look at the loss   100 with probability 0.06 Z =  50 with probability 0.94

Clearly, the 95% CTE for this loss at node (0) is $100. We call this the iterated CTE risk measure, since we iterate the CTE measure backwards from the ﬁnal loss, through the possible paths. The three approaches to the multi-period CTE applied to this example are summarized in Table 1. The re-calculated CTE column assumes the single period CTE is calculated at each node. The accumulated CTE assumes the CTE is calculated at node (0) and carried through to maturity. The Iterated CTE assumes that the risk measure at node (0) is the CTE applied to the two potential CTE values at the next time point.

7 Node Re-calculated Accumulated Iterated CTE(95%) CTE(95%) CTE(95%) (0) 53.60 53.60 100 (1,1) 100 53.60 100 (1,2) 50 53.60 50

Table 1: Risk measure at each node, three CTE approaches to the multi-period situation.

3 The Iterated Risk Measure

3.1 Deﬁnition

It is shown in Wirch and Hardy (1999) that many risk measures can be expressed as expected values of the loss random variable under a suitable change of measure. This is true for both quantile and the CTE risk measures. It is convenient to take advantage of this in expressing the different multi-period risk measures. Let ED[ ] represent the expectation under the distorted probability measure. Different distortions are used for different risk measures. Let Ht,T (L) denote the risk measure at t for a loss L at T > t. Then, allowing for discounting at continuous force of interest r per year, and denoting by

Ft the information on the stock price process up to time t, the risk measure to maturity for an T -year contract at some time t, where 0 ≤ t ≤ n is: h ¯ i −r(n−t)¯ Ht,T (L) = ED L e ¯ Ft (5)

Let Jt,T,k[L] represent the iterated risk measure at t for a loss L at T , assuming iteration at intervals of k years. Assume k n = T − t for some integer n, which represents the number of time steps. We assume equal length of time steps here for convenience, though it is not necessary. In the ﬁnal k-year step of the contract,

JT −k,T,k(L) = HT −k,T (L) (6)

8 Then, for t ≤ T − 2k h ¯ i −r¯ Jt,T,k(L) = ED Jt+k,T,k(L) e ¯ Ft = Ht,t+k(Jt+k,T,k(L)) (7)

So, any single period risk measure can be used to construct a multi-period risk measure by iteration. If we start with a risk measure having some attractive features, such as the CTE, then we retain most of those features after iteration. We use the iterated CTE (ICTE through the remainder of this paper. However, it should be noted that the iterated CTE can be awkward analytically, as it is a function of all the intermediate quantiles. That is,

Ht,t+k(Jt+k,T,k(L)) = ED [Jt+k,T,k(L) |Jt+k,T,k(L) > Qα (Jt+k,T,k(L))] (8)

The two recalculated and accumulated CTE approaches described in Section 2 are also dynamic risk measures. The recalculated risk measure is

RCT E Jt,T (X) = CTE(X|Ft) and the accumulated risk measure is

ACT E Jt,T (X) = CTE(X|F0) accumulated with interest if appropriate.

3.2 Characteristics of the iterated CTE

There are a number of characteristics of one-period risk measures that are commonly considered. The main ones are summarized in the deﬁnition of coherence from Artzner et al (1999). It is appropriate, and suggested in Riedel (2003) and Artzner et al (2003) that a multi-period risk measure should also be tested against the coherence principles, when considered at any speciﬁc time. The coherence principles for a general risk measure H(X) are:

9 1. H(X) ≤ sup(X)

2. H(X) should be positive linear homogeneous; that is, for a > 0, c, H(aX + c) = aH(L) + c

3. H(L) should be subadditive; that is, for two losses X,YH(X +Y ) ≤ H(X)+H(Y ).

The iterated CTE risk measure is a special case of the CTE risk measure. Since the CTE is coherent, so is the iterated CTE, Jt,T,k(L), for any ﬁxed t. The proof of this is simple through backwards induction on t. Fix T and, without loss of generality, let k = 1. Then

JT −1,T,1(L) = HT −1,T (L) which is coherent as it is simply a single period CTE.

Now suppose the coherence criteria are satisﬁed by Jt+1,T,1(L), for t ≤ T −1, and consider

Jt,T,1(L) = Ht,t+1(Jt+1,T,1(L)). Note that Ht,t+1( ) is coherent, as it is a single period CTE, and we will also use the fact that a coherent risk measure is an increasing function of the loss random variable – that is, if X ≥ Y , then Ht,t+1(X) ≥ Ht,t+1(Y )

First, we require that Jt,T,1(L) ≤ sup(L):

Jt,T,1(L) = Ht,t+1 (Jt+1,T,1(L)) ≤ sup (Jt+1,T,1(L)) ≤ sup(sup(L)) = sup(L) as required.

Secondly, we require that Jt,T,1(aL + c) = aJt,T,1(L) + c for a, c > 0. We have

Jt,T,1(aL + c) = Ht,t+1 (Jt+1,T,1(aL + c))

= Ht,t+1 (aJt+1,T,1(L) + c) by inductive assumption

= aHt,t+1 (Jt+1,T,1(L)) + c as H is coherent.

Finally we require subadditivity. For losses L1 and L2 due at time T:

Jt,T,1(L1 + L2) = Ht,t+1 (Jt+1,T,1(L1 + L2))

10 ≤ Ht,t+1 (Jt+1,T,1(L1) + Jt+1,T,1(L2)) by inductive assumption

≤ Ht,t+1 (Jt+1,T,1(L1)) + Ht,t+1 (Jt+1,T,1(L2)) as H is coherent

≤ Jt+1,T,1(L1) + Jt+1,T,1(L2) as required.

So coherence follows for any Jt,T,1, and by a simple change of time unit it is clear that coherence will also be satisﬁed for any Jt,T,k. Riedel(2003) proposes two dynamic axioms in addition to the static coherence axiom; dynamic consistency and relevance. A dynamic risk measure is dynamically consistent if, for two random loss processes X and Y ,

Jt,T,k(X)|Ft = Jt,T,k(Y )|Ft for all Ft

=⇒ Jt−s,T,k(X) = Jt−s,T,k(Y ) for all s, 0 < s ≤ t assuming no cashﬂows associated with the processes in the interval t − s to t. This requirement states that if the risk measures for the losses X and Y are the same at t for every possible state at t, then the risk measures must also be the same for all earlier dates.

A consequence is that Jt−k,T,k(X) cannot be more than the maximum value of Jt,T,k(X), for a consistent risk measure, again assuming no intermediate cash flows. In Artzner et al (2003) dynamic consistency is called time consistency. The iterated risk measure satisfies dynamic consistency since the risk measure at t − k is fully determined by the set of possible values of the measure at t. This is, since Jt−k,T,k(X) is defined as the CTE of the random variables Jt,T,k(X)|Ft, then if two processes have the same values for Jt,T,k(X)|Ft, they must also have the same values for Jt−k,T,k(X), apart from the effects of intermediate cashflows in (t − k, t]. In terms of the binomial example in Section 2, if two different random variables give the same values for the risk measure at nodes (1,1) and (1,2), then a risk measure is consistent if it must give the same value

11 for the risk measure at node (0) for the two random variables. This is clearly true for the iterated CTE, by construction. More formally, we have

Jt,T,1(L) = Ht,t+1(Jt+1,T,1(L))

If for two random variables L1 and L2, Jt+1,T,1(L1) = Jt+1,T,1(L2), for all Ft+1, then clearly

Jt,T,1(L1) = Jt,T,1(L2) also. A dynamic risk measure is relevant if every loss which is not excluded by the current history carries positive risk. The Iterated CTE clearly satisﬁes this criterion. We also suggest that an interesting characteristic of the discrete time multi-period risk measure is that it should be time step increasing. That is, for a liability X due at T , the risk measure at t < T should be an increasing function of the number of time steps that the interval t to T is broken down into. This has similarities to the discrete-monitoring ruin theory problem. If a surplus process is checked for solvency every k years, then the smaller the interval k, the larger the probability of a solvency failure being detected. A particular example of the time-step-increasing characteristic is that if true, then for the iterated and single period risk measures J() and H(), Jt,T,k(L) ≥ Ht,T (L), since H is the single time step version of J. In all the practical applications we have considered, including the lognormal process described below, the iterated CTE is time-step-increasing. However, it is possible to construct examples where this is not true; that is, it is possible for the iterated CTE to be less than the single period CTE in some circumstances. The adherence of the three dynamic risk measures from Section 2 to the criteria of this section may be summarized as follows: Risk Measure Coherent Consistent Relevant Time-Step Increasing ICTE YES YES YES NO . RCTE YES NO YES NO ACTE NO NO NO NO

12 4 The lognormal loss process

In this section we derive the formula for the ICTE for a lognormal loss process, for which analytic results are available. We compare the static CTE with the ICTE, and look at the sensitivity to the time step and to the α parameter.

Suppose Lt follows a geometric Brownian Motion or lognormal process with parameters

µ and σ > 0; at time T = 1 there is a liability L1.

The distribution of Lt+k|Lt, for t + k ≤ 1, is lognormal with parameters µk + log Lt and σ2 k.

The α-quantile of Lt+k|Lt is n √ o Qα(Lt+k|Lt) = Lt exp σ k zα + µ k (9) where the standard normal distribution function at zα is Φ(zα) = α.

The α−CTE of Lt+k|Lt is ³ √ ´ 1 − Φ zα − σ k CTE (L |L ) = L ek (µ+σ2/2) (10) α t+k t t (1 − α)

So, if t = 0 and k = 1 we have the 1-step α−CTE

2 {1 − Φ(z − σ)} H (L ) = L e(µ+σ /2) α (11) 0,1 1 0 (1 − α) Now, suppose the unit of time is split into n = 1/k equal intervals of length k years, n ≥ 1.

Let Jt,T =1,k(L1) denote the iterated CTE at t of the liability due at T = 1, using a time-step of k years. In the ﬁnal k−year time step, the iterated CTE and the 1-step CTE are the same, so that n ³ √ ´o 1 − Φ zα − σ k J (L ) = CTE (L |L ) = L ek (µ+σ2/2) (12) 1−k,1,k 1 α 1 1−k 1−k (1 − α)

13 So, working backwards in k− year steps from T (and using the fact that for a random variable X, and constant c, CTEα(cX) = c CT Eα(X)), we have:

J1−2 k,1,k(L1) = CTEα [CTEα(L1|L1−k)| L1−2 k] (13) n ³ √ ´o 1 − Φ zα − σ k = CTE (L |L ) ek (µ+σ2/2) (14) α 1−k 1−2 k (1 − α)

n ³ √ ´o2 1 − Φ zα − σ k = L e2 k(µ+σ2/2)   (15) 1−2 k (1 − α)

(16)

Similarly

 n ³ √ ´o3 1 − Φ zα − σ k J (L ) = L e3 k(µ+σ2/2)   (17) 1−3 k,1,k 1 1−3 k (1 − α) and continuing backwards to t = 0 we have

n ³ √ ´on  ³ √ ´n 1 − Φ zα − σ k 1 − Φ zα − σ k J (L ) = L ek n(µ+σ2/2) = L e(µ+σ2/2)   (18) 0,1,k 1 0 (1 − α)n 0 1 − Φ(z)

It is relatively easy then to compare the single period CTE, H0,1(L1) from equation (11) and iterated CTE risk measure J0,1,k(L) from equation (18) for the lognormal loss. For a single iteration, that is, n = 1 they are (obviously) the same; for n ≥ 2, the iterated CTE must be greater than the single step CTE, for α > 0. This can be seen as the ratio  ³ √ ´ 1 − Φ zα − σ k   > 1 1 − Φ(z) so that it is an increasing function of n. Note that when α = 0 the CTE is the mean loss, and the iterated expectation is the same

14 14 12 30 10 8 20 6 Ratio of Iterated to Single Period CTE 10 4 2 0

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20

Alpha Number of Iterations

Figure 1: Ratio of iterated CTE to single period CTE for a lognormal loss process.

(µ+σ2/2) as the single step expectation, both being equal to L0 e .

In Figure 1 we show the ratio, J 1 (L1)/H0,1(L1) as a function of α, for n = 5, and as a 0,1, n function of n for α = 0.95. In both cases we let σ = 0.5. Both plots start at 1.0 and increase with α and with n. The sensitivity to the time-step variable is an interesting feature. For this particular example, the iterated 95% CTE for the 10 times iterated CTE term is more than 9 times greater than the single period CTE for the same term. This gives some warning, that the extra security from iteration may come at quite a high price.

15 5 Segregated funds

5.1 Introduction

One area where the CTE risk measure has had an impact is in the determination of economic capital for segregated fund contracts in Canada. These contracts are described in detail in Hardy (2003). In this section we look at the drawbacks of the single period approach, explore how the iterated risk measure might be applied and discuss the implications. The generic policy that we use here is a 10-year single premium contract with a guaranteed minimum death beneﬁt and a guaranteed minimum maturity beneﬁt. The premium is invested into a mutual-type fund, from which a regular charge is deducted to cover expenses. The policyholder receives the fund balance on death or maturity, subject to a minimum of 100% of the initial premium. There is no guarantee on surrender. So, for a premium of $100= G, the guarantee cost to the insurer at maturity, if the policy is still in force, is

L = max(100 − F10, 0) where Ft denotes the value of the fund at t. The required capital for the contract is assumed to be the 95% CTE of the present value of the guarantee liability, net of management charge income. A management charge is deducted monthly from the policyholder’s funds, and a portion of the management charge is available to fund the guarantee cost. This is called the margin oﬀset. In modelling the path of the contract over time, the capital requirements at each valuation date might be determined by simulation. However, this would require two-tiered simulation – that is, for each original path at each projected valuation date, a new set of paths would be required to determine the CTE. So, for a 10-year contract, with the CTE estimated from 1,000 simulations at each projection point, then each projection would re-

16 quire 10,000 simulations, and 1,000 individual paths would require 10,000,000 projections. This is unwieldy, though not entirely infeasible. One way of reducing the number of simulations is to use fewer second tier simulations. The second tier are those used for the CTE calculation. An alternative method is to pre-calculate the CTE at each duration for a number of diﬀerent values for the Fund Value to guarantee ratio. This essentially bunches together the simulations for the purpose of determining the CTE at each date. In other words, at duration t for a contract with full term n > t, we simulate the CTE requirement for, say, Ft/G = 0.3, 0.7, ...2.5. Then when we project the policy forward from the start of the projection (t = 0), the capital requirement at t is determined from rounding the projected value of Ft/G and taking the corresponding CTE requirement from the factors already calculated. This is described in more detail in the next section. With the ICTE the logistical problems using multiple tiered simulation are even greater. For a 10-year contract, rather than two tiered simulation we have 10-tiered simulation. Every iterated CTE calculation at each time point requires multiple-tiered stochastic projection. For each individual projection of a 10-year contract, the iterated CTE calculation would require 100010 projections. This clearly is not feasible. To reduce the dimensions to a manageable undertaking, we again adapt the factor approach. Working backwards from the end of the projection, we summarize the ICTE requirement at each time step with a set

Jt,10,1(PVL) for t = 0, 1, ..., n; and for Ft/G = 0.3, 0.6, ...2.5; where J() is the iterated α-CTE per unit of fund at time t, and PVL is the present value of the future guarantee liability, net of margin oﬀset income. Then, in the projections of the capital requirement we pull from the pre-calculated values of J() the nearest appropriate value according to the projected Fund to Guarantee ratio.

17 5.2 The Recalculated CTE

The current approach to the economic capital requirement is to recalculate the (single period) CTE at each valuation date. We illustrate this method in this section. To avoid the two tier simulation problem, we pre-calculate factors which are applied to the projected fund value at each valuation date (assumed annual) to approximate the CTE requirement. The factors are calculated for values of F/G ranging from 0.3 to 2.5, and are expressed per $100 fund value. For F/G ratios between the values calculated, we interpolate the factors. For ratios below 0.3 we assume capital equal to the discounted guarantee is required. For ratios above 2.5 we use the value at 2.5. Some sample values from the full table are given in Table 2 for the following contract:

Type of contract Guaranteed Minimum Death or Maturity Beneﬁt Term of contract 10 years to maturity Initial Premium $100 Management Charge 3% per year Margin Oﬀset 0.5% per year. Age at maturity 70 Mortality CIA8692L M Agg Ult Lapses 8% per year Risk free rate 6% per year continuously compounded

We use the Regime Switching Lognormal model, with 2 regimes for projecting the fund value. The parameters used are :

µ1 = 0.015 µ2 = −0.02 σ1 = 0.035 σ2 = 0.08 p12 = 0.04 p21 = 0.20

See for example, Hardy (2003) for a more detailed description of this model. These values are chosen to illustrate the process, not to represent any speciﬁc fund value process. To illustrate how these factors are applied, suppose the initial $100 premium has accumulated to $140 after 4 years. The initial guarantee is $100, so the F/G value is 1.4, and

18 Years to Fund Value/ Guarantee Value maturity 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

10 21.2 16.9 13.1 9.7 8.0 6.3 0.9 0 0 9 23.8 19.4 15.4 11.5 8.2 4.1 1.6 0 0 8 27.5 22.6 17.6 13.4 9.2 5.5 2.7 0.2 0 7 30.8 25.9 20.6 16.2 11.3 6.7 4.0 1.1 0 6 35.0 29.3 24.2 18.0 13.4 9.2 4.3 2.2 0 5 40.0 33.2 27.2 21.9 16.0 9.8 5.6 2.7 0.2 4 45.0 38.8 31.9 23.8 18.9 11.5 6.2 2.8 1.2 3 51.4 43.4 37.8 27.4 19.6 11.6 6.6 2.7 1.8 2 57.5 47.6 38.9 29.9 19.9 11.2 5.0 2.6 1.4 1 63.8 52.0 40.4 28.7 17.0 7.0 1.3 0.0 0

Table 2: 95% CTE factors per $100 guarantee value for 10-year GMMB/GMDB contract. there are 6 years left to the final maturity of the contract. The factor from the table is 9.2% of the guarantee. The estimated CTE is then (0.092)(100) = $9.2. For high F/G values, the required capital may be negative, as even in the worst cases the income from margin offset outweighs any outgo. Negative capital requirements are replaced by zero here, to avoid reliance on uncertain future income. We have run 10,000 independent projections of the policy cashflows. Each run involves projecting the fund 10 years, in monthly steps, and at each year end interpolating the figures from the table 2 to determine the capital required. With a 95% CTE, the final capital should be enough to meet the guarantee in all but a few cases. Clearly, adjusting the CTE from year to year dynamically requires minor adjustments. However, the risk manager might be more concerned if there are likely to be major adjustments required to the capital requirements as the contract moves through the term.

19 0.12 0.10 0.08 0.06 0.04 Probability Density Function 0.02 0.0 0 5 10 15 20 Additional Capital, % of Initial Premium

Figure 2: Probability density function of the maximum additional capital requirement for 10-Year GMMB/GMDB, with annual recalculated 95% CTE.

Figure 2 shows the estimated probability density of the maximum additional capital required from this dynamic projection, to meet the year end CTE requirements. This shows that not all the adjustments to the capital are minor. There is an estimated 4.5% probability that, at some point during the term of the contract additional capital will be required of more than 10% of the initial premium – which is more than the initial capital requirement of $9.40%. There is a 30% probability of an additional capital draw at some point of more than 5% of the initial premium. There is no issue here of parameter or model uncertainty, since we project the contract using the same model and parameters as were used to determine the CTE requirement. This example provides an illustration of the need for a dynamic risk measure, ideally able to smooth out the projected volatility in the CTE requirements. An alternative approach, described in Section 2 above, is to use the accumulated CTE. That is, take the $9.70 initial CTE requirement and accumulate until the end of the contract, regardless of the actual path of the fund value. This approach does not recognize the likely regulatory requirements, which in most jurisdictions where equity linked insurance is popular, annual valuation is required, and also ignores the ‘time consistency’ requirement. This says that if the fund value process is favorable, we should be able to release capital once it is clear

20 it will not be required, without waiting until the contract matures. The iterated CTE approach steers a path between these two single-step approaches.

5.3 The Annual Iterated CTE

Using the iterated CTE we break the contract term of 10 years down into separate periods. It seems logical to use annual increments. To overcome the feasibility problem described in Section 5.1 we again use the factor approach to simulating the risk measure at each year end. For the iterated CTE with annual sub-periods, we use repeated 1-year simulations, starting with the final year and working backwards to the first year. Each year assume a fund of $100 at the start of the year. The net cashflows during the year include the margin offset income, the mortality guarantee outgo, and the end year ICTE calculated from the previous year’s projection. We calculate the net present value of all the liabilities during the year for each of, say, 1000 simulations. Then we estimate the iterated CTE at the start of the year. We repeat this for each starting value of the F/G ratio, from 0.3 to 2.5, as above. The iterated CTE in the final year of the contract is the same as the 1-period CTE, so the final row of the table is the same as in Table 2 above. If we compare Table 3 with Table 2, we see the price paid for the additional security of the iterated CTE. At the start of the 10 year contract, the annual iterated 95% CTE is 20.4% of the initial premium, compared with 9.7% for the single period CTE. This increase is substantial. In fact, given that the maximum outgo for this contract is 100% of the initial premium, the iterated CTE is very close to the present value of the maximum outgo, allowing for interest at the risk free rate, and for exits. The objective of the ICTE risk measure is to reduce the downside risk during the course of the contract. In Figure 3 we show some cashflows for the 10-year segregated fund contract, assuming in the top figure that the economic capital is determined using the recalculated CTE, and in the lower figure that the iterated CTE is used. In both cases

21 Years to Fund Value/ Guarantee Value maturity 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

10 21.4 21.2 20.9 20.4 20.0 19.6 19.3 19.0 18.7 9 24.3 24.0 23.6 23.2 22.7 22.3 22.0 21.6 21.0 8 27.5 28.2 26.8 26.3 25.9 25.4 25.0 24.7 24.3 7 31.2 30.9 30.5 30.0 29.5 29.0 28.5 28.1 27.8 6 35.5 35.1 34.7 34.1 33.5 32.9 32.4 31.8 31.1 5 40.4 40.0 39.4 38.7 37.9 36.8 35.8 34.7 33.5 4 46.1 45.6 44.6 43.2 41.3 39.2 36.9 34.6 32.2 3 52.6 51.4 48.7 45.2 41.1 37.0 33.0 29.1 24.7 2 59.7 54.9 48.6 41.6 34.3 27.1 20.7 14.1 9.8 1 63.8 52.0 40.4 28.7 17.0 7.0 1.3 0.0 0

Table 3: 95% ICTE factors per $100 guarantee value for 10-year GMMB/GMDB contract. the heavy lines show the 90% confidence interval for the cashflows in each projection year, estimated by sorting the projections at each year end, and estimating the upper and lower fifth percentiles. Figure 3 demonstrates that using the ICTE does indeed reduce the volatility of the cashflows, and very much reduces the risk of additional capital. The estimated probability of requiring additional capital of more than $10, per $100 initial premium, is estimated at less than 0.2%, compared with 4.8% for the recalculated CTE. However, the price, in terms of carrying additional capital through the term of the contract, is very heavy.

5.4 Mitigating ICTE costs: changing the CTE parameter

Clearly the capital requirement using the ICTE approach are severe at the 95% level. If we reduce the security level the costs will be lower. The time zero requirements for

22 Recalculated CTE 20 10 0 -10 Cashflow, % of Premium -20

0 2 4 6 8 10

Projection Year

Iterated CTE 20 10 0 -10 Cashflow, % of Premium -20

0 2 4 6 8 10

Projection Year

Figure 3: Cash ﬂows using the recalculated or iterated CTE; 90% conﬁdence intervals, with 20 sample paths.

23 CTE parameter Fund Value/ Guarantee Value α 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.95 21.4 21.2 20.9 20.4 20.0 19.6 19.3 19.0 18.7 0.85 21.1 20.7 20.1 19.4 18.8 18.2 17.7 17.2 16.6 0.75 20.7 20.1 19.2 18.2 17.0 16.0 15.1 14.0 12.9 0.65 20.3 19.4 17.9 16.4 14.7 13.0 11.0 9.8 8.0 0.55 19.6 18.1 15.9 13.5 11.1 9.0 6.8 5.0 3.5

Table 4: Starting (time zero) 95% ICTE factors per $100 guarantee value for 10-year GMMB/GMDB contract. different values of α are given in Table 4 below. The table shows that the ICTE for higher values of F/G are more sensitive to the security level changes. The effect of this is that using a lower α reduces the required initial capital a little, for a starting F/G of 1.0, but also allows for a return of the capital sooner if the experience is favorable and the F/G ratio increases. However, reducing security has a price – ultimately the 85% CTE is a weak standard for the final liability. An alternative approach is to change the α standard as the contract moves towards maturity. To illustrate we have set α = 55% for the first year of the contract, increasing by 5% per year to a maximum of 95% for the final two years of the 10 year contract. The results are shown in Table 5 and Figure 4. Figure 4 looks promising. The risk of major additional cashflow requirements are very much lower than we found using recalculated CTEs, and in fact this plot looks similar to the 95% ICTE plot on Figure 3. The volatility of the cashflows is very much mitigated also, and the probability of requiring additional capital of more than 10% of the initial premium is, as with the full 95% ICTE simulations, less than 0.2%. The initial capital requirement of $17.1 for $100 premium using this method is significantly less than the full 95% ICTE, though still substantially more than the re-calculated CTE.

24 Years to Fund Value/ Guarantee Value maturity α 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

10 50% 20.8 19.8 18.5 17.1 15.8 14.6 13.7 12.2 11.1 9 55% 23.7 22.7 21.4 20.0 18.5 17.1 15.9 14.6 13.2 8 60% 26.9 26.0 24.6 23.1 21.4 20.0 18.3 16.8 15.3 7 65% 30.7 29.8 28.2 26.3 24.7 22.8 21.1 18.8 17.4 6 70% 35.0 33.9 32.3 30.0 28.0 25.6 23.1 20.9 19.1 5 75% 39.9 38.7 36.6 33.8 31.2 27.7 24.8 22.7 19.3 4 80% 45.6 43.9 40.8 37.3 33.2 29.4 25.2 21.9 17.8 3 85% 52.1 49.0 44.5 39.8 33.6 28.5 22.8 18.8 15.2 2 90% 59.0 52.7 45.2 38.5 30.3 21.4 15.7 8.3 4.8 1 95% 63.8 52.0 40.4 28.7 17.0 7.0 1.3 0.0 0.0

Table 5: ICTE factors per $100 guarantee value for 10-year GMMB/GMDB contract, linear increasing α.

25 20 10 0 Cashflow, % of Premium -10 -20 0 2 4 6 8 10 Projection Year

Figure 4: Cash ﬂows using the iterated CTE with increasing parameter; 90% conﬁdence intervals, with 20 sample paths.

6 Concluding comments

In this paper we have described how a risk measure may be adapted from single period to multi-period by backwards iteration. Iterating the CTE risk measure gives a dynamic risk measure with some attractive characteristics, including coherence (in the sense of Artzner et al (1999)), and consistency and relevance in the sense of Riedel (2003). Although the ICTE risk measure is awkward analytically in many circumstances, explicit formulae are possible for a lognormal loss process. In other circumstances, including the more realistic and more complex equity-linked example of Section 5 it is possible to estimate the ICTE using stochastic simulation, although some shortcuts are necessary to keep the calculation practicable. We shortcut by using factors for the year end CTE requirements instead of re-simulating these.

26 The ICTE substantially reduces cashflow volatility and the risk of additional interim capital requirements for the segregated fund example, but it does so at the cost of substantial additional capital required through almost all the term of the contract. However, the costs may be reduced whilst keeping the benefit of lower cashflow volatility by using an α parameter that increases as the contract maturity date approaches.

References

Artzner P., Delbaen F., Eber J.-M., Heath D. (1999). Coherent Measures of Risk. Math- ematical Finance 9(3) 203-228. Artzner P., Delbaen F., Eber J.-M., Heath D., Ku H. (2003). Coherent Multiperiod Risk Adjusted Values and Bellman’s Principle. Working Paper, ETH Zurich. Boyle Phelim P. and Vorst, Ton (2003) VaR behaving Badly. Working paper, University of Waterloo. O’Connor M. (2002) Hedging liability risks, micro and macro approaches: risk management of variable annuities. Presented to the 2002 Investment Actuary symposium. www.soa.org/conted/investment symposium/oconnor.pdf Riedel F. (2003). Dynamic Coherent Risk Measures. Working paper. Stanford University Department of Economics. Segregated Funds Task Force (SFTF) (2002) Report of the Task Force on Segre- gated Fund Investment Guarantees Canadian Institute of Actuaries; available from http://www.actuaries.ca/publications/2002/202012e.pdf Wang, T. (2000). A Class of Dynamic Risk Measures. Working paper, University of British Columbia. Wirch J. L. and Hardy M. R. (1999). A synthesis of risk measures for capital adequacy. Insurance: Mathematics and Economics. 25 337-347.